Let V=Mn(F) be the space of all nxn matrices over F; define TA=(1/2)(A+transpose(A)) for A in V.
Verify that T is not only a linear operator on V, but is also a projection.
A is a projection when A squared=A.
The Attempt at a Solution
I don't see how this works since clearly (1/2)(A+transpose(A)) squared does not equal (1/2)(A+transpose(A)) for all matrices.
What am I doing wrong?